Problem 59 - College Algebra and Calculus: An Applied Approach

Question

The number of a certain type of bacteria increases according to the model \(P(t)=100 e^{0.01896 t}\) where \(t\) is time (in hours). (a) Find \(P(0)\). (b) Find \(P(5)\). (c) Find \(P(10)\). (d) Find \(P(24)\).

Step-by-Step Solution

Answer

\n(a) \(P(0) = 100\)\n(b) \(P(5) \approx 109.58\)\n(c) \(P(10) \approx 120.27\)\n(d) \(P(24) \approx 174.72\)

Step 1: Evaluate P(0)

Substitute \(0\) for \(t\) in the equation \(P(t) = 100e^{0.01896t}\) to find \(P(0)\). After doing this, we see that \(P(0) = 100e^{0.01896*0} = 100\).

Step 2: Evaluate P(5)

To find \(P(5)\), substitute 5 for \(t\) in the equation \(P(t) = 100e^{0.01896t}\). This yields \(P(5) = 100e^{0.01896*5}\). The value can be calculated using a calculator, and it gives approximately 109.58.

Step 3: Evaluate P(10)

Substitute 10 for \(t\) in the equation \(P(t) = 100e^{0.01896t}\) to find \(P(10)\). This gives \(P(10) = 100e^{0.01896*10}\). Using a calculator, this yields approximately 120.27.

Step 4: Evaluate P(24)

Substitute 24 for \(t\) in the equation \(P(t) = 100e^{0.01896t}\) to find \(P(24)\). So, \(P(24) = 100e^{0.01896*24}\). Using a calculator, this yields approximately 174.72.

Key Concepts

Bacteria Population Model

Exponential growth models are incredibly useful for understanding how populations grow, especially in biology. In our bacteria population model, the formula that describes this growth is given by \[ P(t) = 100e^{0.01896t} \] where \(P(t)\) represents the population of bacteria at time \(t\) measured in hours. This equation shows that the growth rate is constant, which is what makes it an exponential function.​

• The initial population is 100 bacteria. This is derived from the coefficient in front of the exponential function.
• The growth rate is 0.01896 per hour. This rate is constant and is expressed in the exponent of \(e\), the base of natural logarithms.

Understanding exponential functions is crucial because they describe a process where the growth rate, rather than the total amount, increases continuously over time. This model indicates that each hour, the number of bacteria increases by a consistent percentage, leading to rapid growth as time progresses.

Time in Exponential Functions

The role of time in exponential functions such as our bacteria population model is central to the change observed over continuous periods. Time \(t\) is the independent variable that determines how far along an exponential function progresses in terms of growth.

• For \(t = 0\), the population remains at its initial value, as any number raised to the power of zero equals 1.
• As \(t\) increases, the exponentiation in the equation \(e^{0.01896t}\) becomes significant, thereby increasing the value of \(P(t)\).

In most real-world applications, time is a key factor in accurately predicting how quickly or slowly exponential processes, like bacterial growth, evolve. The exponential function takes into account this progression and reflects a real-world dynamic where populations multiply rapidly as time progresses.

Calculating Growth at Specific Times

To calculate population growth at specific times using an exponential model, such as our bacteria population model, one substitutes the desired time into the function. Let's see how this works with specific examples.

- At \(t = 0\): By substituting 0 for \(t\) in \[ P(t) = 100e^{0.01896\times 0} = 100 \] we find that the initial population at time zero is simply the initial given value of 100 bacteria.- At \(t = 5\): Substituting 5 for \(t\), we have \[ P(5) = 100e^{0.01896\times 5} \] which calculates to approximately 109.58 bacteria.- At \(t = 10\): The equation becomes \[ P(10) = 100e^{0.01896\times 10} \] which results in about 120.27 bacteria.- Finally, at \(t = 24\): \[ P(24) = 100e^{0.01896\times 24} \] calculates to approximately 174.72 bacteria.
These calculations show how population size increases exponentially over time. This demonstrates the rapid and substantial growth that defines exponential functions.